The functorial source problem via dimension data

TL;DR

This paper addresses the functorial source problem for certain automorphic representations by analyzing dimension data of S-subgroups, introducing quasi root systems as a key tool.

Contribution

It develops a new approach to determine the functorial source using distinction and linear independence of dimension data, and introduces quasi root systems for studying S-subgroups.

Findings

Proves distinction and linear independence among dimension data of S-subgroups.

Defines and utilizes quasi root systems to analyze S-subgroups.

Provides new results on the functorial source problem for specific automorphic representations.

Abstract

For an automorphic representation $\pi$ of Ramanujan type, there is a conjectural subgroup $\mathcal{H}_{\pi}$ of the Langlands L-group $^{L}G$ associated to $\pi$, called the {\it functional source} of $\pi$. The functorial source problem proposed by Langlands and refined by Arthur intends to determine $\mathcal{H}_{\pi}$ via analytic and arithmetic data of $\pi$. In this paper, we consider the functorial source problem of automorphic representations of a split group, a unitary group, or an orthogonal group which do not come from endoscopy and have minimal possible ramification. In this setting, $\mathcal{H}_{\pi}$ must be an S-subgroup of $^{L}G$. We approach the functorial source problem by proving distinction and linear independence among dimension data of S-subgroups. Nice results along this direction are shown in this paper. We define a notion of quasi root system and use it as the key tool for studying S-subgroups and their dimension data.